Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities

References

[1] V. I. Afanas’ev, A limit theorem for a critical branching process in a random environment, Diskret. Mat. 5 (1993), no. 1, 45–58. 10.1515/dma.2001.11.6.587Search in Google Scholar

[2] V. I. Afanas’ev, A new limit theorem for a critical branching process in a random environment, Discrete Math. Appl. 7 (1997), no. 5, 497–513. 10.1515/dma.1997.7.5.497Search in Google Scholar

[3] G. Alsmeyer, D. Buraczewski and A. Iksanov, Null recurrence and transience of random difference equations in the contractive case, J. Appl. Probab. 54 (2017), no. 4, 1089–1110. 10.1017/jpr.2017.54Search in Google Scholar

[4] G. Alsmeyer and A. Iksanov, A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks, Electron. J. Probab. 14 (2009), no. 10, 289–312. 10.1214/EJP.v14-596Search in Google Scholar

[5] G. Alsmeyer and A. Iksanov, Recurrence and transience of random difference equations in the critical case, preprint (2021), http://arxiv.org/abs/2105.04994. Search in Google Scholar

[6] G. Alsmeyer, A. Iksanov and U. Rösler, On distributional properties of perpetuities, J. Theoret. Probab. 22 (2009), no. 3, 666–682. 10.1007/s10959-008-0156-8Search in Google Scholar

[7] K. B. Athreya and S. Karlin, On branching processes with random environments. I. Extinction probabilities, Ann. Math. Statist. 42 (1971), 1499–1520. 10.1214/aoms/1177693150Search in Google Scholar

[8] K. B. Athreya and P. E. Ney, Branching Processes, Grundlehren Math. Wiss. 196, Springer, New York, 1972. 10.1007/978-3-642-65371-1Search in Google Scholar

[9] M. Babillot, P. Bougerol and L. Elie, The random difference equation X n = A n ⁢ X n - 1 + B n in the critical case, Ann. Probab. 25 (1997), no. 1, 478–493. 10.1214/aop/1024404297Search in Google Scholar

[10] K. A. Borovkov and V. A. Vatutin, Reduced critical branching processes in random environment, Stochastic Process. Appl. 71 (1997), no. 2, 225–240. 10.1016/S0304-4149(97)00074-4Search in Google Scholar

[11] D. Buraczewski, E. Damek and T. Mikosch, Stochastic Models with Power-Law Tails, Springer Ser. Oper. Res. Financ. Eng., Springer, Cham, 2016. 10.1007/978-3-319-29679-1Search in Google Scholar

[12] Y. S. Chow and H. Teicher, Probability Theory. Independence, Interchangeability, Martingales, 3rd ed., Springer Texts Statist., Springer, New York, 1997. 10.1007/978-1-4612-1950-7Search in Google Scholar

[13] K. Fleischmann and V. A. Vatutin, Reduced subcritical Galton–Watson processes in a random environment, Adv. in Appl. Probab. 31 (1999), no. 1, 88–111. 10.1239/aap/1029954268Search in Google Scholar

[14] J. Geiger and G. Kersting, The survival probability of a critical branching process in random environment, Theory Probab. Appl. 45 (2000), no. 3, 518–526. 10.4213/tvp491Search in Google Scholar

[15] J. Geiger, G. Kersting and V. A. Vatutin, Limit theorems for subcritical branching processes in random environment, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 4, 593–620. 10.1016/S0246-0203(02)00020-1Search in Google Scholar

[16] C. M. Goldie and R. A. Maller, Stability of perpetuities, Ann. Probab. 28 (2000), no. 3, 1195–1218. 10.1214/aop/1019160331Search in Google Scholar

[17] A. Iksanov, Renewal Theory for Perturbed Random Walks and Similar Processes, Probab. Appl., Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-49113-4Search in Google Scholar

[18] G. Kersting and V. Vatutin, Discrete Time Branching Processes in Random Environment, Math. Stat. Ser., Wiley-ISTE, London, 2017. 10.1002/9781119452898Search in Google Scholar

[19] H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math. 131 (1973), 207–248. 10.1007/BF02392040Search in Google Scholar

[20] F. C. Klebaner, U. Rösler and S. Sagitov, Transformations of Galton–Watson processes and linear fractional reproduction, Adv. in Appl. Probab. 39 (2007), no. 4, 1036–1053. 10.1239/aap/1198177238Search in Google Scholar

[21] M. V. Kozlov, The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment, Theory Probab. Appl. 21 (1976), no. 4, 791–804. 10.1137/1121091Search in Google Scholar

[22] A. Lindo, Some computational aspects of Markov processes, PhD thesis, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden, 2016. Search in Google Scholar

[23] S. Sagitov and A. Lindo, A special family of Galton–Watson processes with explosions, Branching Processes and Their Applications, Lect. Notes Stat. 219, Springer, Cham (2016), 237–254. 10.1007/978-3-319-31641-3_14Search in Google Scholar

[24] V. Vatutin and E. Dyakonova, Reduced branching processes in random environment, Mathematics and Computer Science. II (Versailles 2002), Trends Math., Birkhäuser, Basel (2002), 455–467. 10.1137/S0040585X97983845Search in Google Scholar

[25] V. A. Vatutin and E. E. Dyakonova, Critical branching processes in a random environment: probabilities of extinction at a fixed time, Diskret. Mat. 9 (1997), no. 4, 100–126. Search in Google Scholar

[26] W. Vervaat, On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab. 11 (1979), no. 4, 750–783. 10.2307/1426858Search in Google Scholar

[27] M. P. W. Zerner, Recurrence and transience of contractive autoregressive processes and related Markov chains, Electron. J. Probab. 23 (2018), Paper No. 27. 10.1214/18-EJP152Search in Google Scholar